The value of service rate flexibility in an queue with admission control

We consider a single-server queueing system with admission control and the possibility to switch dynamically between k increasing service rate values with the service cost rate being convex in the service rate. We explore the benefit due to service rate flexibility on the optimal profit and the admission thresholds, when service payment is made upon customer's admission. We formulate a Markov Decision Process model for the problem of joint admission and service control considering both discounted and average expected profit maximization and show that the optimal policy has a threshold structure for both controls. Regarding the benefit due to flexibility, we show that it is increasing in system length and that its effect on the admission policy is to increase the admission threshold. We also derive a simple approximate condition between the admission reward and the relative cost of service rate increase, so that the service rate flexibility is beneficial. We finally show that the results extend to the corresponding model where service payment is made at the end of each service completion and differences regarding the benefit due to service flexibility with respect to the original model are pursued numerically.